Mathematical Sciences: Smooth Structures of Algebraic Surfaces

小行星9400729 自从Simon唐纳森十年前的基础工作以来，规范理论已经成功地应用于光滑四维流形，特别是复杂曲面的研究。 本项目的主要内容是利用Simon唐纳森和Dieter Kotschick提出的光滑不变量，通过研究复曲面上的稳定向量丛及其模空间，以及利用Jun Li和Robert Friedman最近的工作，研究复曲面的微分拓扑与复几何之间的相互作用. 特别是，调查希望研究唐纳森不变量简单连接代数曲面平凡几何属（包括复杂的射影平面）。 这个项目是在规范理论领域，它与其他数学领域，如拓扑，代数几何，甚至数学物理有很大的相互作用。 进步可以采取许多不同的形式。 令人惊讶的是，最近一些物理学家（特别是爱德华维滕）在超对称杨-米尔斯理论方面的工作，在复杂曲面的微分拓扑学中产生了许多数学见解。 很有可能，对这个项目中所讨论的一些代数几何和代数拓扑的更深入的理解将反过来阐明量子物理学家社区所关心的问题。 ***

9400729 Qin Since the fundamental work of Simon Donaldson about a decade ago, gauge theory has been very sucessfully applied to study smooth 4-manifolds, especially complex surfaces. The main concern of this project is to study the interplay between the complex geometry and the differential topology of complex surfaces, by utilizing smooth invariants introduced by Simon Donaldson and Dieter Kotschick, by investigating stable vector bundles and their moduli spaces on complex surfaces, and by using recent work of Jun Li and Robert Friedman. In particular, the investigator wishes to study Donaldson invariants for simply connected algebraic surfaces with trivial geometric genus (including the complex projective plane). This project is in the field of gauge theory, which interacts heavily with other areas of mathematics such as topology, algebraic geometry, and even mathematical physics. Progress could take many different forms. Surprisingly, recent work in supersymmetric Yang-Mills theory by some physicists (notably Edward Witten) has led to many mathematical insights in the differential topology of complex surfaces. Quite possibly, a deeper understanding of some of the algebraic geometry and algebraic topology at issue in this project will in turn shed light on problems of concern to the community of quantum physicists. ***